The roots of this discussion lies in the classical work of Halmos [1]} in which he established the Beurling's theorem for vector-valued Hardy spaces. He used the Wold-von Neumann decomposition theorem for isometries, which states that an isometric operator \(V\)  on \(\mathcal {H}\)  always admits a decomposition \(V= V|_{\mathcal {H}_0}\oplus V|_{\mathcal {H}_1}\)  where, \(\mathcal {H}_0,\mathcal {H}_1\)  are reducing subspaces for \(V\)  such that \(V|_{\mathcal {H}_{0}}\)  is a unitary, \(V|_{\mathcal {H}_1}\)  is a pure isometry and \(\mathcal {H}= \mathcal {H}_{0} \oplus \mathcal {H}_{1}\) . Furthermore, the subspaces \(\mathcal {H}_0, \mathcal {H}_1\)  have the following precise structure
\(\mathcal {H}_{0} = \bigcap _{m=1}^{\infty } V^m \mathcal {H}; \quad \mathcal {H}_1 = \bigoplus _{m=0}^{\infty } V^{m} \mathcal {W}(V).\) 
